Since the salt is molten, the only ions are $\ce{Ag+}$ and $\ce{NO3-}$, and both of them can only do a reduction reaction. Who goes to the anode? Also, there is no acid in the solution — only the salt — therefore the following reaction cannot occur, right?

$$\ce{ NO3- + 4H+ + 3e- -> NO + H2O}$$

So, what happens? The only possible reaction is the reduction of the $\ce{Ag+}$ to $\ce{Ag (s)}$. There is literally nothing else.

1 Answer
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In principle, a nitrate reduction in the presence or silver ions only would have to give silver oxide; but this decomposes between 200 and 300°C. So, perforce, you would ultimately get elemental silver from electrolyzing the nitrate melt.